Problem: s involving definite integrals (algebraic) AP.CALC: CHA‑4 (EU), CHA‑4.D (LO), CHA‑4.D.1 (EK), CHA‑4.D.2 (EK), CHA‑4.E (LO), CHA‑4.E.1 (EK) Google Classroom Facebook Twitter Email You might need: Calculator Problem The water level of a strait is changing at a rate of $\dfrac{3}{2}\sin\left(2-\dfrac{t}{2}\right)$ centimeters per hour (where $t$ is the hours since midnight). By approximately how many centimeters does the water level change between $t=1$ and $t=4$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $1.4$ (Choice B) B $1.5$ (Choice C) C $2.8$ (Choice D) D $3.0$
Answer: Letting $w(t)$ be the water level $t$ hours after midnight, we are given that $w'(t)=\dfrac{3}{2}\sin\left(2-\dfrac{t}{2}\right)$. We want to find $w(4)-w(1)$. According to the Fundamental Theorem of Calculus, $\begin{aligned} w(4)-w(1)&=\int_{1}^{4} w'\left(t\right)dt \\\\ &=\int_{1}^{4}\dfrac{3}{2}\sin\left(2-\dfrac{t}{2}\right)dt \end{aligned}$ $\int_{1}^{4}\dfrac{3}{2}\sin\left(2-\dfrac{t}{2}\right)dt\approx2.79$ In conclusion, between $t=1$ and $t=4$ the water level increased by approximately $2.8$ centimeters.